First-order Boundary Layer Research Articles

A process-based model for sediment transport under various wave and current conditions

The purpose of this study is to investigate the capability of a newly developed process-based model for sediment transport under a wide variety of wave and current conditions. The model is based on the first-order boundary layer equation and the sediment advection-diffusion equation. In particular, a modified low Reynolds number k-ɛ model is coupled to provide the turbulence closure. Detailed model verifications have been performed by simulating a number of laboratory experiments, covering a considerable range of hydrodynamic conditions such as sinusoidal waves, asymmetric waves and wave-current interactions. The model provides satisfactory numerical results which agree well with the measured results, including the time-averaged/dependent sediment concentration profiles and sediment flux profiles, as well as the time series of concentration at given elevations. The observed influences of wave orbital velocity amplitude, wave period and sediment grain size are correctly reproduced, indicating that the fundamental physical mechanisms of those processes are properly represented in the model. It is revealed that the present model is capable of predicting sediment transport under a wide range of wave and current conditions, and can be used to further study the morphodynamic processes in real coastal regions.

Modeling wave–current bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-e model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.

Effect of moving wall on the three-dimensional separation of an incompressible fluid in front of an obstacle

The problem of the laminar boundary layer formed on the surface of a semiinfinite plate with a perpendicular semi-infinite circular cylinder in a uniform steady incompressible flow normal to the leading edge is considered. Near its sharp edge the plate has a stationary part and, located at a finite distance further downstream, a part of the surface moving downstream at a constant velocity. The first-order boundary layer equations are solved numerically by an implicit finite-difference method. The effect of the moving wall on the variation of the dimensions of the separation zone ahead of the obstacle over a broad range of the governing parameters and flow characteristics is investigated. The flow in the laminar boundary layer on the surface of a plate ahead of such an obstacle was calculated in [1, 2] without motion of the wall. Data on the structure of the separated flow are given in [3].

Free convection from a semi-infinite vertical surface bounded by a horizontal wall in a porous medium

A study of the steady free convection flow along a semi-infinite vertical flat plate which is placed in a saturated porous medium and at an arbitrary distance d above a horizontal wall is performed. The first- and second-order boundary layer equations and outer inviscid flow equations are studied to find the effects of large, but finite, values of the Rayleigh number. Results are obtained for the two cases in which the flat plate is (i) isothermal and (ii) uniform flux. It is found that the first-order boundary layer solutions overestimate the local Nusselt number except for the uniform flux plate for distances along the plate less than about 0.4 d.

Unsteady laminar incompressible second-order boundary-layer flow at a three-dimensional stagnation point

The unsteadely laminar incompressible second-order boundary-layer flow at the stagnation point of a three-dimensional body has been studied for both nodal and saddle point regions. The effects of mass transfer and Prandtl number have been taken into account. The equations governing the flow have been solved numerically using an implicit finite-difference scheme. It has been found that the parameter characterizing the unsteadiness in the velocity of the free stream, the nature of the stagnation point, the mass transfer and Prandtl number strongly affect the second-order skin friction and heat transfer. The overall skin friction becomes less due to second-order effects but the heat transfer has the opposite behaviour. For large injection, the second-order skin-friction and heat-transfer results prevail over the first-order boundary layer results whereas for the case of large suction the behaviour is just the opposite.

Darcian free convective flow about an impermeable horizontal surface bounded by a vertical wall

The problem of steady free convection in a porous medium adjacent to a horizontal impermeable heated surface, with wall temperature T ̄ W = T ̄ 0 + A x ̄ λ (0 ⩽ λ < 2) for x ̄ ⩾ 0 and T ̄ = T 0 for − d ⩽ x ̄ < 0 and a vertical wall with temperature T ̄ W = T ̄ 0 for x ̄ = − d, y ̄ > 0 , is investigated by the method of matched asymptotic expansions. The first—and second—order boundary layer equations and the outer inviscid flow equations are studied to find the effects of large, but finite values of the Rayleigh number. The small parameter in the perturbation series is found to be the inverse one-third power of the Rayleigh number. It is found that λ = 1 2 is a critical value and therefore results are presented in detail for the two cases of λ = 0 and 1. It is found that the first-order boundary layer solutions overestimate the local Nusselt number except when λ > 1 2 and only then for short distances along the heated surface. Further, the horizontal velocity of the fluid on the heated surface is overestimated by the first-order boundary layer theory except for λ < 1 2 and for x̄ sufficiently small.

Unsteady compressible second-order boundary layers at the stagnation point of two-dimensional and axisymmetric bodies

The unsteady laminar compressible boundary-layer flow over two-dimensional and axisymmetric bodies at the stagnation point with mass transfer has been studied for all second-order boundary layer effects when the basic potential flow admits self-similarity. The solutions for the governing equations are obtained by using an implicit finite-difference scheme. Computations have been carried out for different values of the parameters characterizing the unsteadiness in the free stream velocity, wall temperature, mass transfer rate and variable gas properties. The results are found to be strongly affected by the unsteadiness in the free stream velocity. For large injection rates the second-order boundary layer effects may prevail over the first-order boundary layer, but reverse is true for suction. The wall temperature and the variation of the density-viscosity product across the boundary layer appreciably change the skin-friction and heat-transfer rates due to second-order boundary-layer effects.

Compressible second-order boundary layers for three-dimensional stagnation point flows with mass transfer

All the second-order boundary-layer effects have been studied for the steady laminar compressible 3-dimensional stagnation-point flows with variable properties and mass transfer for both saddle and nodal point regions. The governing equations have been solved numerically using an implicit finite-difference scheme. Results for the heat transfer and skin friction have been obtained for several values of the mass-transfer rate, wall temperature, and also for several values of parameters characterizing the nature of stagnation point and variable gas properties. The second-order effects on the heat transfer and skin friction at the wall are found to be significant and at large injection rates, they dominate over the results of the first-order boundary layer, but the effect of large suction is just the opposite. In general, the second-order effects are more pronounced in the saddle-point region than in the nodal-point region. The overall heat-transfer rate for the 3-dimensional flows is found to be more than that of the 2-dimensional flows.

A similarity solution for combined forced and free convection flow over a horizontal plate

The effect of buoyancy forces on the steady, laminar, plane flow over a horizontal plate is investigated within the framework of a first-order boundary layer theory, taking into account the hydrostatic pressure variation normal to the plate. An exact similarity solution is given for the case of a wall temperature that is inversely proportional to the square root of the distance from the leading edge. Remarkably, such a solution does not exist if the buoyancy parameter is smaller than a certain critical value, which is negative (decelerated flow) but not yet small enough to satisfy the separation criterion of vanishing shear stress at the wall. Although the wall temperature is different from the free-stream temperature, there is no local heat transfer at the wall except in the singular point at the leading edge. The total heat transfer is finite, independent of the plate length, and is calculated by applying the heat flux equation. The displacement thickness is also given. It is negative if the plate is heated sufficiently strongly.

Second Approximation to the Laminar Boundary-Layer Flow over a Parabola

The method of matched asymptotic expansions is applied to the problem of a parabolic cylinder in a uniform incompressible flow at high Reynolds numbers. The equations and the boundary conditions for the first-order boundary layer, the second-order outer flow and the second-order boundary layer are formulated in terms of the parabolic co-ordinate system. The equations are solved numerically by means of finite-difference approximation. The results for skin friction and surface pressure are compared with the available numerical solutions of the full Navier-Stokes equations. The application of the technique of rational fraction to the two-term asymptotic expansion for skin friction shows that a further approximation will give a better result.

Reacting flow as an example of a boundary layer under singular external conditions

An inviscid flow with infinite gradients at the wall poses a problem for the accompanying boundary layer that is fundamentally different from the conventional one of bounded gradients. It turns out that in both cases the external gradients do not affect the first-order boundary layer, but the unbounded gradients generate second-order corrections that are of a lower order in Reynolds number than the conventional ones. The present singularity in stagnation-point reacting flow is of an algebraic type, and the boundary-layer corrections that it generates are proportional to non-integral powers of the Reynolds number, with exponents that vanish with the rate of reaction. The present example clarifies such matters as the matching of boundary layer and singular inviscid flow, the structure and decay of the new corrections, and their ranking in comparison with the conventional second-order effects. Numerical computations illustrate the problem and give quantitative results in a few selected cases.